A Sharp Small-Coefficient Variant of Khintchine's Inequality and the Sharp $\pi \mathrm{/}2$ Theorem

We prove a refined quadratic normal approximation for the first absolute moment of normalized weighted Rademacher sums with bounded maximal coefficients. For any weight vector $w\in {\mathbb{R}}^n$ satisfying $\mathrm{\parallel }w{\mathrm{\parallel }}_2=1$ and $\mathrm{\parallel }w{\mathrm{\parallel }}_{\mathrm{\infty }}\le \beta$ with sufficiently small $\beta >0$, we establish the uniform error bound $\mathrm{\mid }\mathbb{E}\mathrm{\mid }{\sum }_{i=1}^n{w}_i{X}_i\mathrm{\mid }-\sqrt{2\mathrm{/}\pi }\mathrm{\mid }=O({\beta }^2)$ over all admissible weight configurations. Our proof combines zero-bias Stein’s method and refined small-ball probability estimates to exploit symmetry cancellation and control the non-smooth residual of the absolute-value test function. An explicit extremal construction further verifies the optimality of this quadratic convergence rate. As an application, we establish an asymptotically sharp refinement of the Friedgut–Kalai–Naor (FKN) theorem for Boolean functions, also known as the sharp $\pi \mathrm{/}2$ theorem, characterizing the level-1 Fourier energy for functions deviating far from dictatorships.